Help with purple fringing and backlighting

I took a trip to Italy last month and for the most part was very pleased with the quality of the photos taken with my P850. I had initially intended on becoming more familiar with more of the 'technical' features of the P850 and taking more pictures in manual modes instead of just automatic. Unfortunately, work was crazy prior to my trip and then once I was there I didn't want to waste time fiddling with my camera.

I found 2 'types' of pictures which I had problems with. One involving taking pictures of sunlight coming through domes (shot of Pantheon is below). I had seen bits about 'purple fringing' on this forum, but didn't pay much attention to it since I had never encountered it. Is there any way to fix this? I don't have any good photo-editing software (just EasyShare and Picasa).

The second 'type' of picture I had problems with was backlit shots. We climbed to the top of both St. Peter's in Vatican City and Il Duomo in Florence, and all of the shots of us with scenic background behind us came out fairly washed out. Is there any way to get the background darker without changing the foreground?

Any help would be greatly appreciated. For the most part I was very happy with my pictures, but I wish these ones had come out better. And yes, my New Year's resolution is to better understand the capabilities of my camera